<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">f</font> = <a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">h</font> = <a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<a NAME="E1:84"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T101">RELAT_1:101</a></i>;<br/>
<i><font color="Green">E23</font></i>: <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> 
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:84"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T103">RELAT_1:103</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T40">RFUNCT_2:40</a></i>
<br/>.= 
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="txt" href="sin_cos6.html#E2:84"><i><font color="Green">E23</font></i></a>, <a class="txt" href="sin_cos6.html#E31"><i><font color="Green">Lemma66</font></i></a>, <a class="ref" href="extens_1.html#T1">EXTENS_1:1</a></i>
;<br/>
<a NAME="E3:84"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="partfun2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos6.html#T1" target="_self">Th1</a>, <a class="ref" href="sin_cos6.html#T15" target="_self">Th15</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E4:84"/><i><font color="Green">E31</font></i>: 
 <a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T73">SIN_COS:73</a></i>;<br/>
<a NAME="E5:84"/><b>then </b><i><font color="Green">E32</font></i>: 
<a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T33" target="_self">Th33</a>, <a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T32" target="_self">Th32</a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K19">sin</a> ,<font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T32" target="_self">Th32</a>, <a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="ref" href="sin_cos.html#T73">SIN_COS:73</a></i>
;<br/><b>hence </b><a NAME="E3:84_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a>, <a class="ref" href="comptrig.html#T27">COMPTRIG:27</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:84"/><b>then </b><i><font color="Green">E35</font></i>: 
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T29" target="_self">Th29</a>, <a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos6.html#T31" target="_self">Th31</a>, <a class="ref" href="sin_cos6.html#T33" target="_self">Th33</a>, <a class="ref" href="fdiff_2.html#T48">FDIFF_2:48</a></i>;<br/>
<a NAME="E8:84"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos6.html#K1" target="_self">arcsin</a> 
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T15" target="_self">Th15</a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E9:84"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span> holds <br/><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E10:84"/><i><font color="Green">E106</font></i>: 
 <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="sin_cos6.html#T36" target="_self">Th36</a>, <a class="ref" href="fdiff_1.html#D7">FDIFF_1:def 7</a></i>;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">x</font> = <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>;<br/>
<b>assume </b><a NAME="E11:84"/><i><font color="Green">E107</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E12:84"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E13:84"/><i><font color="Green">E109</font></i>: 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E14:84"/>
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font>
 
;<br/>
<a NAME="E15:84"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> &amp; <a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, <a class="ref" href="sin_cos6.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E16:84"/><b>then </b><i><font color="Green">E110</font></i>: 
<a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<i><font color="Green">E111</font></i>:  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T33" target="_self">Th33</a>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K19">sin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T32" target="_self">Th32</a>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K19">sin</a> ,<span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T32" target="_self">Th32</a>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<a href="sin_cos.html#K22">cos</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#T73">SIN_COS:73</a></i>
<br/>.= 
 <a href="sin_cos.html#K24">cos</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K3" target="_self">arcsin</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D23">SIN_COS:def 23</a></i>
<br/>.= 
 <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="sin_cos6.html#T28" target="_self">Th28</a></i>
;<br/>
<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos6.html#K1" target="_self">arcsin</a> ,<font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="fdiff_1.html#K2">`|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="fdiff_2.html#T16">FDIFF_2:16</a></i>
<br/>.= 
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K1" target="_self">arcsin</a>  <a href="relset_1.html#K8">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,1</span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>,<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#T35" target="_self">Th35</a>, <a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#T29" target="_self">Th29</a>, <a class="ref" href="sin_cos6.html#T30" target="_self">Th30</a>, <a class="ref" href="sin_cos6.html#T31" target="_self">Th31</a>, <a class="ref" href="sin_cos6.html#T33" target="_self">Th33</a>, <a class="ref" href="sin_cos6.html#T34" target="_self">Th34</a>, <a class="ref" href="sin_cos6.html#T45" target="_self">Th45</a>, <a class="ref" href="sin_cos6.html#T46" target="_self">Th46</a>, <a class="txt" href="sin_cos6.html#E10:84"><i><font color="Green">E106</font></i></a>, <a class="ref" href="fdiff_2.html#T48">FDIFF_2:48</a></i>
;<br/>


</div>
